nLab icon

Definition

For 2-categories

Let $C,D$ be 2-categories and $F,G:C\to D$ be 2-functors. An icon [Lack 2010, cf. Rem. ] $\alpha \colon F\to G$ consists of:

1. The assertion that $F$ and $G$ agree on objects.

2. For each 1-morphism $u \colon x\to y$ in $C$, a 2-morphism $\alpha_u:F(u) \to G(u)$ in $D$.

   (Note that this only makes sense because $F$ and $G$ agree on objects, so that $F(u)$ and $G(u)$ are parallel.)

3. For each 2-morphism $\mu \colon u\to v$ in $C$, we have $\alpha_v . F(\mu) = G(\mu).\alpha_u$.

4. For each object $x$ of $C$, $\alpha_{1_x}$ is an identity

   (modulo the unit constraints of $F$ and $G$, if they are not strict functors).

5. For each composable pair $x\overset{u}{\to} y \overset{v}{\to} z$ in $C$, we have $\alpha_v {*} \alpha_u = \alpha_{v u}$

   (modulo the composition constraints of $F$ and $G$, if they are not strict 2-functors).

Remark

If $D$ is a strict 2-category (or at least strictly unital), then an icon is identical to an oplax natural transformation whose 1-morphism components are identities.

In general, there is a bijection between icons and such oplax natural transformations, obtained by pre- and post-composing with the unit constraints of $D$. The name “ICON” derives from this correspondence: it is an Identity Component Oplax Natural-transformation.

Applications

Icons have technical importance in the theory of 2-categories. For instance, there is no 2-category (or even 3-category) of 2-categories, functors, and lax or oplax transformations (even with modifications), but there is a 2-category of 2-categories, functors, and icons. (In fact, this 2-category is the 2-category of algebras for a certain 2-monad.)

Additionally, if monoidal categories are regarded as one-object 2-categories, then monoidal functors can be identified with 2-functors, and monoidal transformations can be identified with icons.

Icons are also used to construct distributors in the context of enriched bicategories.

Relation to pseudo double categories

A bicategory can be viewed as a pseudo double category whose tight-cells are trivial. An icon is then precisely a transformation of oplax functors of pseudo double categories. See Paré for details.

References

An early observation that restricting to icons allows one to form a bicategory of bicategories and lax functors appears in:

• Aurelio Carboni, and Robert Rosebrugh. Lax monads: Indexed monoidal monads, Journal of Pure and Applied Algebra 76.1 (1991): 13-32.

The terminology “ICON” was introduced in:

• Stephen Lack: Icons, Appl. Categ. Structures. 18 3 (2010) 289-307 [arxiv:0711.4657, doi:10.1007/s10485-008-9136-5]

• Richard Garner, Mike Shulman, Section 3.8 of Enriched categories as a free cocompletion, Adv. Math. 289 (2016), 1–94

• Robert Paré. Composition of modules for lax functors. Theory and Applications of Categories 27.16 (2013): 393-444.

• Niles Johnson, Donald Yau, Section 4.6 of: 2-Dimensional Categories, Oxford University Press 2021 (arXiv:2002.06055, doi:10.1093/oso/9780198871378.001.0001)